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Visual Differential Geometry and Forms: A Mathematical Drama in Five Acts
An inviting, intuitive, and visual exploration of differential geometry and forms
Visual Differential Geometry and Forms fulfills two principal goals. In the first four acts, Tristan Needham puts the geometry back into differential geometry. Using 235 hand-drawn diagrams, Needham deploys Newton's geometrical methods to provide geometrical explanations of the classical results. In the fifth act, he offers the first undergraduate introduction to differential forms that treats advanced topics in an intuitive and geometrical manner.
Unique features of the first four acts include: four distinct geometrical proofs of the fundamentally important Global Gauss-Bonnet theorem, providing a stunning link between local geometry and global topology; a simple, geometrical proof of Gauss's famous Theorema Egregium; a complete geometrical treatment of the Riemann curvature tensor of an n-manifold; and a detailed geometrical treatment of Einstein's field equation, describing gravity as curved spacetime (General Relativity), together with its implications for gravitational waves, black holes, and cosmology. The final act elucidates such topics as the unification of all the integral theorems of vector calculus; the elegant reformulation of Maxwell's equations of electromagnetism in terms of 2-forms; de Rham cohomology; differential geometry via Cartan's method of moving frames; and the calculation of the Riemann tensor using curvature 2-forms. Six of the seven chapters of Act V can be read completely independently from the rest of the book.
Requiring only basic calculus and geometry, Visual Differential Geometry and Forms provocatively rethinks the way this important area of mathematics should be considered and taught.
Visual Differential Geometry and Forms fulfills two principal goals. In the first four acts, Tristan Needham puts the geometry back into differential geometry. Using 235 hand-drawn diagrams, Needham deploys Newton's geometrical methods to provide geometrical explanations of the classical results. In the fifth act, he offers the first undergraduate introduction to differential forms that treats advanced topics in an intuitive and geometrical manner.
Unique features of the first four acts include: four distinct geometrical proofs of the fundamentally important Global Gauss-Bonnet theorem, providing a stunning link between local geometry and global topology; a simple, geometrical proof of Gauss's famous Theorema Egregium; a complete geometrical treatment of the Riemann curvature tensor of an n-manifold; and a detailed geometrical treatment of Einstein's field equation, describing gravity as curved spacetime (General Relativity), together with its implications for gravitational waves, black holes, and cosmology. The final act elucidates such topics as the unification of all the integral theorems of vector calculus; the elegant reformulation of Maxwell's equations of electromagnetism in terms of 2-forms; de Rham cohomology; differential geometry via Cartan's method of moving frames; and the calculation of the Riemann tensor using curvature 2-forms. Six of the seven chapters of Act V can be read completely independently from the rest of the book.
Requiring only basic calculus and geometry, Visual Differential Geometry and Forms provocatively rethinks the way this important area of mathematics should be considered and taught.
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Displaying 1 - 8 of 8 reviews
September 10, 2021
Part 5 (the only one I've read) is far and away the best intuitive explanation of differential forms I've come across. Like Visual Complex Analysis I wouldn't recommend reading this without a more formal book as a companion.
Parts 1-4 are Riemannian geometry, I probably won't read them.
Maybe worth noting that the prose is overwrought and kind of annoying, but it basically only shows up in the intro sections so whatever.
Update: I was not nearly positive enough in my initial review. This is getting added to my list of best math textbooks ever. Still needs a formal book to go with it though.
Update 2: I glanced through the other 4 sections. Overall they felt much less valuable, probably because the topics covered already have easily accessible geometric intuitions. I don't think getting an even more visual than usual explanation of curvature is very helpful, especially when it comes at the cost of rigor.
Parts 1-4 are Riemannian geometry, I probably won't read them.
Maybe worth noting that the prose is overwrought and kind of annoying, but it basically only shows up in the intro sections so whatever.
Update: I was not nearly positive enough in my initial review. This is getting added to my list of best math textbooks ever. Still needs a formal book to go with it though.
Update 2: I glanced through the other 4 sections. Overall they felt much less valuable, probably because the topics covered already have easily accessible geometric intuitions. I don't think getting an even more visual than usual explanation of curvature is very helpful, especially when it comes at the cost of rigor.
September 25, 2025
Putting the geometry back in Differential Geometry
If you're a mathematician, you are repeatedly reminded that there are vast reaches of mathematics, important and beautiful, of which you personally know almost nothing. (I am, at least.) For me one of those areas is differential geometry. Oversimplifying a bit, differential geometry is the study of curved spaces. It's important to physicists mainly because, according to Einstein (and a hundred years of experimental results) gravity is caused by the curvature of space and time. I have studied the math of gravity, so I am not entirely ignorant of differential geometry. I didn't really understand it, though. It was just a bunch of formulas that didn't really make much sense.
Thus, when I recently discovered that Tristan Needham had published a textbook of differential geometry (VDG below), I was intrigued. I was intrigued because many years ago I read Needham's other book, Visual Complex Analysis (VCA in what follows), and it was great!
Needham begins his preface with this quote,
Needham is a man who loves, loves, loves geometry, more than almost anything. In his Acknowledgements he claims to love his wife Mary and his twin daughters more, but it is obvious that the competition is close.
Now for a confession. The right way to read a math textbook is to work through it carefully, making sure you understand the proofs and working the exercises. Reading a math text without working the exercises is about as effective as visiting a gym and reading the instructions for the machines and the free weights, without actually using them. Working the exercises takes a long time, and I didn't want to spend so much of my life on this book. Thus I decided to read it like a novel. Needham's Visual Complex Analysis can be read that way (it's the pictures that do it), so I had hopes that this would still be worthwhile. Those hopes were mostly justified.
Needham's account of differential geometry is divided, like a play, into five acts. The first three acts present what I will call classical (that is, pre-Riemann) differential geometry, which is mostly about the geometry of two- dimensional spaces. Examples would be the surface of the Earth. The Earth is a three-dimensional object, but its surface is pretty close to a a two-dimensional space. It's a curved two-dimensional space. Acts 1-3 mostly explore the geometry of two-dimensional spaces. This was almost entirely new to me, and it was the greatest win of the "read it like a novel" gamble. I now understand classical differential geometry far better than I did before reading
Act IV extends the discussion to spaces of more than two dimensions. This was Bernhard Riemann's great contribution to differential geometry, which became the mathematical basis of Einstein's theory of gravity. No one knows how Riemann did it. In spaces of more than two dimensions, the curvature at each point in space is a complicated object now called the "Riemann tensor." Every modern treatment of the Riemann tensor (including the ones I had studied previously) depends on an idea called "parallel transport" that would not be invented until 1917, long after Riemann's death in 1866, and even after Einstein's publication of his theory of gravity. Needham's Act IV was not a noticeable improvement on the previous presentations of general relativity that had left me unsatisfied in the past. It felt as if he had given in to the Devil's bargain described in the quote -- it was mostly algebra, without much of Needham's visual geometric intuition. There probably is a better way to do this, because Riemann did it. But Riemann is possibly the best mathematician of all time, so I can't really fault Needham for not finding it.
This is especially true because he redeems himself in Act V, "Forms," where he presents Elie Cartan's theory of forms and uses it for a visual, geometric presentation of differential geometry in more than two dimensions. Cartan's forms were almost entire new math to me. Act V was the greatest failure of my "read it like a novel" strategy. I benefitted, but I would have learned a lot more if I had worked the exercises. Perhaps I will, one day.
I'm not sure if I have conveyed how clearly Needham's voice sounds throughout VDG. He's a Star Trek fan* who is deeply moved by mathematical beauty†.
*The index entry for "Star Trek" speaks for itself
Star Trek
Captain Kirk, 191, 482
Dr. McCoy, 38
forms proof of (“Star Trek phaser”) metric curvature formula, 452
geometric proof of (“Star Trek phaser”) metric curvature formula, 266
Mr. Spock, 191, 476, 482
NCC-1701, 435
(“Star Trek phaser”) metric curvature formula, 38
The City on the Edge of Forever, 38
The Doomsday Machine, 191
†He describes the flash of insight that led him to a geometrical proof of one of the most important theorems of classical differential geometry in a footnote, "the startling, unanticipated flash of clarity—in the Sierra mountains, surrounded by pristine snow—was one of the happiest moments of my life."
Blog review.
If you're a mathematician, you are repeatedly reminded that there are vast reaches of mathematics, important and beautiful, of which you personally know almost nothing. (I am, at least.) For me one of those areas is differential geometry. Oversimplifying a bit, differential geometry is the study of curved spaces. It's important to physicists mainly because, according to Einstein (and a hundred years of experimental results) gravity is caused by the curvature of space and time. I have studied the math of gravity, so I am not entirely ignorant of differential geometry. I didn't really understand it, though. It was just a bunch of formulas that didn't really make much sense.
Thus, when I recently discovered that Tristan Needham had published a textbook of differential geometry (VDG below), I was intrigued. I was intrigued because many years ago I read Needham's other book, Visual Complex Analysis (VCA in what follows), and it was great!
Needham begins his preface with this quote,
Algebra is the offer made by the devil to the mathematician. The devil says: “I will give you this powerful machine, and it will answer any question you like. All you need to do is give me your soul: give up geometry and you will have this marvellous machine.”... the danger to our soul is there, because when you pass over into algebraic calculation, essentially you stop thinking: you stop thinking geometrically, you stop thinking about the meaning.That felt exactly like my experience with gravity.
-- Sir Michael Atiyah
Needham is a man who loves, loves, loves geometry, more than almost anything. In his Acknowledgements he claims to love his wife Mary and his twin daughters more, but it is obvious that the competition is close.
Now for a confession. The right way to read a math textbook is to work through it carefully, making sure you understand the proofs and working the exercises. Reading a math text without working the exercises is about as effective as visiting a gym and reading the instructions for the machines and the free weights, without actually using them. Working the exercises takes a long time, and I didn't want to spend so much of my life on this book. Thus I decided to read it like a novel. Needham's Visual Complex Analysis can be read that way (it's the pictures that do it), so I had hopes that this would still be worthwhile. Those hopes were mostly justified.
Needham's account of differential geometry is divided, like a play, into five acts. The first three acts present what I will call classical (that is, pre-Riemann) differential geometry, which is mostly about the geometry of two- dimensional spaces. Examples would be the surface of the Earth. The Earth is a three-dimensional object, but its surface is pretty close to a a two-dimensional space. It's a curved two-dimensional space. Acts 1-3 mostly explore the geometry of two-dimensional spaces. This was almost entirely new to me, and it was the greatest win of the "read it like a novel" gamble. I now understand classical differential geometry far better than I did before reading
Act IV extends the discussion to spaces of more than two dimensions. This was Bernhard Riemann's great contribution to differential geometry, which became the mathematical basis of Einstein's theory of gravity. No one knows how Riemann did it. In spaces of more than two dimensions, the curvature at each point in space is a complicated object now called the "Riemann tensor." Every modern treatment of the Riemann tensor (including the ones I had studied previously) depends on an idea called "parallel transport" that would not be invented until 1917, long after Riemann's death in 1866, and even after Einstein's publication of his theory of gravity. Needham's Act IV was not a noticeable improvement on the previous presentations of general relativity that had left me unsatisfied in the past. It felt as if he had given in to the Devil's bargain described in the quote -- it was mostly algebra, without much of Needham's visual geometric intuition. There probably is a better way to do this, because Riemann did it. But Riemann is possibly the best mathematician of all time, so I can't really fault Needham for not finding it.
This is especially true because he redeems himself in Act V, "Forms," where he presents Elie Cartan's theory of forms and uses it for a visual, geometric presentation of differential geometry in more than two dimensions. Cartan's forms were almost entire new math to me. Act V was the greatest failure of my "read it like a novel" strategy. I benefitted, but I would have learned a lot more if I had worked the exercises. Perhaps I will, one day.
I'm not sure if I have conveyed how clearly Needham's voice sounds throughout VDG. He's a Star Trek fan* who is deeply moved by mathematical beauty†.
*The index entry for "Star Trek" speaks for itself
Star Trek
Captain Kirk, 191, 482
Dr. McCoy, 38
forms proof of (“Star Trek phaser”) metric curvature formula, 452
geometric proof of (“Star Trek phaser”) metric curvature formula, 266
Mr. Spock, 191, 476, 482
NCC-1701, 435
(“Star Trek phaser”) metric curvature formula, 38
The City on the Edge of Forever, 38
The Doomsday Machine, 191
†He describes the flash of insight that led him to a geometrical proof of one of the most important theorems of classical differential geometry in a footnote, "the startling, unanticipated flash of clarity—in the Sierra mountains, surrounded by pristine snow—was one of the happiest moments of my life."
Blog review.
June 26, 2022
As others on our site here have commented correctly, this text needs to be read after, and along with, more formal, topical discussions. That having been said, with the appropriate background and reference aids at hand, the reader will be taken through a well written and, at times, extraordinarily clear presentation.
October 30, 2022
A delightful little textbook for differential geometry
Differential geometry done through deep intuition guiding pictures. I really enjoyed it, and it feels like a lot of it bedded down quite deeply in my mind. Sometimes lacked in the clarity of arguments you expect from a maths textbook (I didn't quite know what had been shown using what, for example), but broadly excellent!
Good if you want to understand, rather than just serve as a reference of truth.
Differential geometry done through deep intuition guiding pictures. I really enjoyed it, and it feels like a lot of it bedded down quite deeply in my mind. Sometimes lacked in the clarity of arguments you expect from a maths textbook (I didn't quite know what had been shown using what, for example), but broadly excellent!
Good if you want to understand, rather than just serve as a reference of truth.
February 14, 2025
I stumbled upon this book almost by accident, and now I have fallen head over heals in love with differential geometry. Needham's geometric approach to explaining and proving astounding mathematic theorems is both intuitive and satisfying and is sequenced such that every new discovery is both obvious and exciting. Needham's commentary is gripping while still lighthearted and even humorous at times. My biggest critiques are that I wish we spent more time with higher-dimensional manifolds and Act V about differential forms felt more like an afterthought to me than the jaw-dropping conclusion it was established as. However, I think my lack of experience studying this field played a large role in these critiques, and the sheer sense of wonder this book brought me more than made up for them.
September 2, 2023
Good source for intuition when needed.
April 17, 2022
I enjoyed the first three sections of the book immensely. As someone for whom geometric insight is lacking, I thought that the book did an excellent job helping me to build the tools to perform geometric analysis. Part 4 was inaccessible to me, as I was not patient enough to sit with a companion text to help me better understand the underlying concepts and new notation schema.
I will likely go through this book again, and I am sure I will pick up the Visual Complex Analysis book and work through that (though, admittedly, not for some time).
I will likely go through this book again, and I am sure I will pick up the Visual Complex Analysis book and work through that (though, admittedly, not for some time).
March 12, 2023
Loved this book. Read it in between jobs in 2021. Part of my nightly read before bed. Got a better understanding of curvature, hyperbolic geometry, and forms.
Displaying 1 - 8 of 8 reviews